Homoclinically expansive actions and a Garden of Eden theorem for harmonic models

Abstract

Let $\Gamma$ be a countable Abelian group and $f\in \mathbb{Z}[\Gamma ]$, where $\mathbb{Z}[\Gamma ]$ denotes the integral group ring of $\Gamma$. Consider the Pontryagin dual $X_f$ of the cyclic $\mathbb{Z}[\Gamma ]$-module $\mathbb{Z}[\Gamma ]/\mathbb{Z}[\Gamma ]f$ and suppose that $f$ is weakly expansive (e.g., $f$ is invertible in ${\ell }^1(\Gamma )$, or, when $\Gamma$ is not virtually $\mathbb{Z}$ or ${\mathbb{Z}}^2$, $f$ is well-balanced) and that $X_f$ is connected. We prove that if $\tau :X_f\to X_f$ is a $\Gamma$-equivariant continuous map, then $\tau$ is surjective if and only if the restriction of $\tau$ to each $\Gamma$-homoclinicity class is injective. We also show that this equivalence remains valid in the case when $\Gamma ={\mathbb{Z}}^d$ and $f\in \mathbb{Z}[\Gamma ]=\mathbb{Z}[u_1,u_1^{-1},\dots ,u_d,u_d^{-1}]$ is an irreducible atoral polynomial such that its zero-set $Z(f)$ is contained in the image of the intersection of $[0,1{]}^d$ and a finite union of hyperplanes in ${\mathbb{R}}^d$ under the quotient map $\R^d \to \T^d$ (e.g., when $d\ge 2$ such that $Z(f)$ is finite). These two results are analogues of the classical Garden of Eden theorem of Moore and Myhill for cellular automata with finite alphabet over $\Gamma$.

arXiv admin note: text overlap with arXiv:1706.06548